The stati energy at high orders from NSPT simulationsThe stati energy at high orders fromNSPT simulationsClemens Bauer and Gunnar Bali, University of RegensburgAntonio Pineda, UAB Bar elonaSTRONGNET 2010, Cyprus

The stati energy at high orders from NSPT simulations1 Numeri al Sto hasti Perturbation Theory2 Renormalons3 Summary

The stati energy at high orders from NSPT simulationsNumeri al Sto hasti Perturbation TheorySto hasti Quantizationalternative way of al ulating expe tation values in Eu lidian FTParisi, Wu (1981)1 additional, titious time oordinate τ :φ(x) → φ(x, τ)2 Langevin equation

∂

∂τφ(x, τ) = − δS

δφ(x, τ)+ η(x, τ)3 expe tation values via

O[φ] = limT→∞

1

T

∫ T

0dτ

∫

D [η] P [η] O [φ]

The stati energy at high orders from NSPT simulationsNumeri al Sto hasti Perturbation TheorySto hasti Quantizationalternative way of al ulating expe tation values in Eu lidian FTParisi, Wu (1981)1 additional, titious time oordinate τ :φ(x) → φ(x, τ)2 Langevin equation

∂

∂τφ(x, τ) = − δS

δφ(x, τ)+ η(x, τ)3 expe tation values via

O[φ] = limT→∞

1

T

∫ T

0dτ

∫

D [η] P [η] O [φ] = 〈O[φ]〉

The stati energy at high orders from NSPT simulationsNumeri al Sto hasti Perturbation TheoryNumeri al Sto hasti Perturbation Theory (NSPT)Sto hasti Quantization implemented numeri ally for Latti e GTDi Renzo et al. (1994)1 Langevin equation for Gauge elds:∂

∂τUµ(x, τ) = −i

(

∇x,µSG,L[U ] + taηaµ(x, τ)

)

Uµ(x, τ)2 Gauge elds as perturbative expansionU = 1+β− 1

2 U (1)+β−1U (2)+· · ·+β−M

2 U (M); β− 1

2 =g0√2Nc

→ al ulation ost ∝ M2

→ NSPT heaper than diagrammati LPT at high orders M !

The stati energy at high orders from NSPT simulationsRenormalonsperturbation theory: generi observable R as seriesR =

∑

n

cnαn oe ients cn grow fast:cn

n→∞∼ Kann!nb

→ perturbative expansion does not onverge,at best it is an asymptoti seriesRenormalons ertain pattern of fa torial growth of cne.g. arise when inserting bubble hains in Feynman diagramssmall and large momentum behaviour origin:UV and IR renormalons

The stati energy at high orders from NSPT simulationsRenormalonstool: summation via Borel transform and Borel integralR ∼

∞∑

n=0

rnαn+1 =⇒ B[R](t) =

∞∑

n=0

rntn

n!

R =

∞∫

0

dt e−t/α B[R](t)

→ behaviour of perturbative expansion R di tated by:the losest singularityu = −tβ0to the origin of its Borel transform

→ leading renormalon

The stati energy at high orders from NSPT simulationsRenormalonsFirst observable: The plaquette〈P 〉 ≡ 〈1−1

3trUP 〉 =

∑

n

cnαn+π2

36CGG(α) a4 〈α

πGG〉latt+ O(a6) ontains gluon ondensate 〈GµνGµν〉leading IR renormalon at u = 2

→ al ulate plaquette to high orders with NSPT→ early works with NSPT, e.g.Di Renzo et al. hep-th/9502095, 8 loops,Di Renzo et al. hep-lat/0011067, 10 loops,Rakow hep-lat/0510046, 16 loops

The stati energy at high orders from NSPT simulationsRenormalonsat leading order:notation :

∞∑

n=0

rnαn+1,

LOPlaq = limn→∞

rn

rn−1=

11

8πn

The stati energy at high orders from NSPT simulationsRenormalons

0 5 10 15 20 25 300

0.5

1

1.5

2

2.5

3

n

r n/(n*

r n−1)

Plaquette 43x8 PBCLO

Plaq

The stati energy at high orders from NSPT simulationsRenormalonsse ond observable:Polyakov LoopP(~x) =

1

3Tr

L4−1∏

τ=0

U4(~x, τ)

The stati energy at high orders from NSPT simulationsRenormalonsse ond observable:Polyakov LoopP(~x) =

1

3Tr

L4−1∏

τ=0

U4(~x, τ)gives a ess to stati quark self energy Vself :Vself = lim

L4→∞

(

− 1

L4ln 〈P〉

)

,

Vself =∑

V(n)self α

n

The stati energy at high orders from NSPT simulationsRenormalonsVself of an innitely heavy quark is linearly UV divergentleading UV Renormalon appears already at u = 1/2

→ renormalon should emerge four times faster as for plaquette→ at leading order

notation :∞∑

n=0

rnαn+1,LOV = limn→∞

rn

rn−1= 4 LOPlaq

The stati energy at high orders from NSPT simulationsRenormalons

0 5 10 15 20 25 300

0.5

1

1.5

2

2.5

3

3.5

4

4.5

n

r n/(n*

r n−1)

Plaquette 43x8 PBCLO

Plaq

Vself

43x8 PBC

LOV

The stati energy at high orders from NSPT simulationsRenormalonsPeriodi boundary onditions (PBC): zero momentum modesappear→ NSPT treatment: subtra t them after ea h update

The stati energy at high orders from NSPT simulationsRenormalonsPeriodi boundary onditions (PBC): zero momentum modesappear→ NSPT treatment: subtra t them after ea h updateTwisted Boundary Conditions

Uµ(x + Lν) = ΩνUµ(x)Ω†νat least two dire tions must be twisted onstant twist matri es Ων yield

ΩµΩν = ηΩνΩµ, η ∈ Z(N)for instan e η = e2πi/3 for SU(3)2 options for implementation: either expli it hoi e of Ων ,or phase fa tors for ertain plaquettes at orners of twistedplanes

The stati energy at high orders from NSPT simulationsRenormalonsEe ts of Twisted Boundary Conditions:1 elimination of zero modes → no subtra tion needed2 momenta kν in twisted dire tions quantized as if the SU(N)gauge elds lived on a latti e of size L × N instead of L:kν =

2π

LNnν, ν = twisted dire tion,

2π

Lnν, ν = periodi dire tion.

→ redu tion of nite size ee ts

The stati energy at high orders from NSPT simulationsRenormalons

0 2 4 6 8 10 120.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

n

r n/(n*

r n−1)

Plaquette 43x8 PBC

Plaquette 63x12 TBC

The stati energy at high orders from NSPT simulationsRenormalonsfrom Trottier et al. hep-lat/0110051:

The stati energy at high orders from NSPT simulationsRenormalons

0 2 4 6 8 10 120.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

n

r n/(n*

r n−1)

Plaquette 43x8 PBC

Plaquette 63x12 TBCV

self 43x8 PBC

Vself

63x12 TBC

The stati energy at high orders from NSPT simulationsRenormalons

0 2 4 6 8 10 120.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

n

r n/(n*

r n−1)

Plaquette 43x8 PBC

Plaquette 63x12 TBCV

self 43x8 PBC

Vself

63x12 TBC

Vself

83x12 PBC

The stati energy at high orders from NSPT simulationsSummary SummaryNSPT is a powerful tool for high-order al ulations→ renormalon physi sVself renormalon should emerge a lot earlier than the usual andidate from the gluon ondensateVself severely ae ted by nite size ee ts when using periodi boundary onditionstwisted boundary onditions learly redu e nite size ee ts:distin t ratio urves at moderate latti esfurther simulations on larger latti e volumes needed to de ideon renormalon existen e